Short FormHypothesis TestingOne-sample t-test for a correlation coefficientCORRELATION COEFFICIENTMeasure of closeness of linear relationship between 2 variables—ρ for population; r for sample 22yyxxyyxxriiiioryyxxxyniniiininiiinininiiiiiLLLnyynxxnyxyxr*1212121211 1 Quantitative measure of the dependence between 2 variables—best if both variables are normally distributedExample: A Patient Satisfaction Test has been in use in hospitals. We want to evaluate a new Short Form of the test. Both were administered to a sample of patients. The following results were obtained:Patient Short Form(X)Standard(Y)1 40 452 55 613 60 594 65 825 70 806 73 767 80 908 85 1029 90 9810 97 10011 100 11412 105 11113 111 115Summary Statistics:38395777104113315987103122,YX,YY,XXiiiiii969029075703384655276923603176935392384655271311337771041310311598713113310313839522....*..,,*,rRelationship of correlation coefficient toregression coefficient:Note: 221122andXXYYr*bbYYXXriiiiFor the example,b1 is a rescaled version of r with the scale factor the ratio of the standard deviation of Y to the standard deviation of X025011310311598713113310313839521.,*,bProperties of r1. Pure number2. Ranges from –1 to + 13. If b is positive, r is positive; if b is negative, r isnegative; if b = 0, r = 04. If r > 0, positive correlation5. If r < 0, negative correlation6. If r = 0, not correlated7. r does not change if X (or Y) changes units or if X (or Y) is multiplied by a constantUse the regression coefficient when you want to predict one variable from a second variable. Use the correlation coefficient when you want to describe the linear relationship between 2 variables.Hypothesis TestingOne-sample t-test for a correlation coefficientHo: = 0HA: 0 = 0.0521,22~12ntrnrtReject Ho if t > tn-2, 0.975 or t < tn-2, 0.025For the example, reject Ho if t > 2.201 or t < -2.201006132471214396912139691222.....rnrtReject
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